This is the fourth day of a long term project started in previous lessons (Checkerboard squares, Squares in a row, and Checkerboard poster) in which students have determined the number of squares on an 8 by 8 checkerboard, a 6 by 6, a 4 by 4. In previous lessons students have already begun using data tables to record the different size squares possible and how many of each there are. Most students have completed the total numbers and some have already started excitedly talking about some patterns they have noticed. Students will begin creating their posters and today is a workday for them.
This warmup is designed to help them notice, describe, and represent patterns in the data as they work on creating their poster describing a checkerboard problems.
The purpose of the first two problems 8-1^2 and (8-1)^2 is twofold. Firstly it is a reminder of the order of operations which we reviewed in earlier lessons in this unit (Writing the rules, Writing the rules part 2, 4 Fours poster, Out of order ). When we go over these two I ask what difference the parentheses made in how we solved the problem. It is also intended to help them recognize the pattern of decreasing square numbers (64+49+36....) in the checkerboard problem. In tomorrow's lesson I will make reference to this pattern again when we start generalizing the pattern.
The next two problems (8^2+7^2 and 8^2+7^2+6^2) may help them see another pattern when they compare posters of different sizes. Namely that you can calculate the total number of squares on the next larger size square checkerboard just by adding the next square number. I am trying to get them to really start paying attention to patterns as well as learning to articulate and describe them thouroughly. Students in my district have trouble articulating their own ideas in math, perhaps partly due to their lack of confidence that their ideas have any value since they are not in the top math classes. In addition it is possible that students have been taught to take a passive role in the classroom.
The last problem in the warmup is an In/Out table that follows the rule n^2+1. The numbers I provide them are not in order, because I don't want them to say "you add 2 more than you did before". I find that giving the numbers out of order helps them see the multiplicative rule more easily. When I go over this one I really incorporate wait time, because some students will start to notice the pattern as other students fill in the blanks as we go. Before I ask for "what comes out when n goes in" I make sure they take a moment to discuss what is being done to each number, have them share out, and then ask "so, for any number, n, that goes in...."
Students will spend the remainder of the class period creating their poster in pairs or trios. They received the requirements in a previous lesson (Checkerboard poster) and I have two sample posters posted in front of the room.
Today is the day they find out who their partner or team is and which size checkerboard I have asigned them. In the last lesson (checkerboard poster) I made this decision based on where they were in their solution. Students who were having trouble with the larger checkerboard were asigned the 4 by 4. Students who seemed to take a little longer or who had trouble staying focused were asigned the 4 by 4 or the 6 by 6. Students who were totally on track or who had talked about patterns they had noticed were asigned the 8 by 8. Once the posters are complete (it will probably take two days) I want each math family group to have a variety of posters so they can start generalizing the patterns, so I try to asign an equal number of each size. I asign a few more of the 8by 8s, because these are likely the students who will choose my other option, which is to make a poster for a size (9 by 9) checkerboard they have not worked with yet. They are only required to have 2 of the 5 required elements on their poster if they choose this option.
Before they begin I distribute a handout (look for patterns in your poster) for them to work on for the next two days. They know they are being graded on their teamwork as well, so I go over what kinds of ways they can contribute to their project. While only one person might be working on the poster at a time the partner can be looking for and talking about patterns that might be emerging, or helping to make decisions about how and where to show a certain element on the poster, etc. I also encourage them to take out and use their checkerboard asignment (how many squares with a table).
Because they have a fairly long time to work they are allowed to get up and get supplies as they need them (markers, rulers, pencils, erasers) or to get up and look at the sample posters.
As I circulate I am looking and listening for students who may be stuck. I am also making sure they understand and are refering to the required elements. I ask which element they are working on, how they decided to represent it and why, and where they plan to put the other elements. As I see and hear students working really collaboratively I will highlight it for the class. "This team is noticing a connection between the size of the checkerboard and the numbers in the table", "I like the way this team is really making decisions together", "This team has found a really cool way to show the 3rd required element, you should come take a look", etc.
At the end of class I tell them that they will have the following day to complete the project, but if they want to plan to work together after school they can take it with them.